A-1.4 Modulo Division

In modulo arithmetic, modulo division is different from regular numeric division. Strictly speaking, there is no such thing as modulo division because the modulo operation itself is a form of division that returns only the remainder. In practice, one uses “modulo division” to mean multiplying by a modular inverse when it exists, i.e., when $\gcd <!--\; FUNCTION\; APPLICATION\; -->(a,q)=1$. Modulo division of $b$ by $a$ modulo $q$ is equivalent to computing the modular multiplication $b\cdot a^{-1}modq$. The result of modulo division is different from that of numeric division, because modulo division always gives an integer (a residue modulo $q$) (as it multiplies two integers modulo $q$), whereas numeric division gives a real number. The inverse of an integer modulo $q$ can be computed using the extended Euclidean algorithm (YouTube tutorial).